The main motivation for this thesis is to develop and improve numerical tools and methods that help further our comprehension of the volcanic plumbing system and its dynamics.

The commonly used standard elastic model predicts solutions of dyke shape, thickness, over-pressure and fracturing criteria that do not always ﬁt natural observations. In the ﬁrst part of the thesis, we want to test whether other host-rock rheologies leads to more realistic dykes. We examine three different rheologies: 1) elasticity with pressure dependent elastic moduli, 2) elastoplasticity with plastic failure in regions of high shear stresses and 3) viscoelasticity to describe ductile ﬂow of rocks by creeping mechanisms in regions of high temperature. Solu-tions from the three tested models give dykes with more rectangular shapes relative to solutions computed from the model of linear elasticity. In addition, the calculated magma pressure for an intrusion of a given thickness is reduced for all tested rheologies. Greatest differences with the linear elastic solution are given by the elastoplastic model, in which computed magma over-pressures are lower than elastic solutions by a factor of 2 to 10. Computed overpressures from this type of rheological model are approximately of the same order than natural magma over-pressures estimated by other methods (1 to 5 MPa). For the model of dyke propagation, the incorporation of brittle failure mechanisms in regions of high stresses strongly affects the dyke propagation criteria because of the energy dissipated by frictional sliding and fracturing in the large process zone located at the intrusion tips.

The second part of the thesis deals with the rotation of crystals suspended in magmatic ﬂows. We proceed by coupling the rotation dynamics equation of elongated particles, with the Navier-Stokes equation of large-scale ﬂows. The results of the model are ﬁrst extensively tested for simple ﬂows with known analytical solutions. Results show a perfect ﬁt between both numerical and analytical solutions. Additionally, the numerical methods are applied to more complex ﬂow ﬁelds that relate to realistic systems of magma circulation. Results show that elongated crystals mainly align in the direction of ﬂow in convergent systems (e.g. magma ﬂowing from a large reservoir inside a conduit or from a deﬂating magma chamber). However, the pattern of crystal orientation in divergent ﬂows (e.g. magma ﬂowing from a conduit into a large reservoir or in an inﬂating magma chamber) does not align in the direction of the ﬂow but instead is globally oriented sub-parallel to the maximum principal strain.